基于微分变换策略的全离散间断Galerkin方法
High Order Discontinuous Galerkin Methods with Differential Transform Strategy
摘要: 在本文中，针对一维双曲守恒律方程，我们基于微分变换策略提出了一种高阶全离散间断Galerkin方法。与传统间断Galerkin方法相比较，该方法的主要特点为存储量较小，在时间上能够到达任意高阶精度。与ADER方法相比较，避免了复杂的Cauchy-Kowalewski步骤，同时程序编写简单。数值结果表明该方法具有高阶精度，针对间断解保持高分辨率。
Abstract: In this paper, based on the differential transform strategy, we introduce high order full discrete discontinuous Galerkin methods for one-dimensional hyperbolic conservation laws. Compared with the standard discontinuous Galerkin methods, the current methods enjoy the following advantages including low storage as well as keeping arbitrary high order accuracy in time. In comparison with the ADER methods, the resulting methods avoid the complicated Cauchy-Kowalewski procedure and the coding is easy at the same time. In addition, numerical results also indicate that the resulting methods enjoy genuine high order accuracy for smooth solutions, and keep steep discontinuity transition.
文章引用：于海燕, 王秀芳, 李刚. 基于微分变换策略的全离散间断Galerkin方法[J]. 应用数学进展, 2018, 7(5): 574-583. https://doi.org/10.12677/AAM.2018.75068

参考文献

[1] Reed, W.H. and Hill, T.R. (1973) Trigangular Mesh Methods for the Neutron Transportation Equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory.
[2] Cockburn, B. and Shu, C.-W. (1989) TVB Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws for Conservation Laws II: General Framework. Mathematics of Computation, 52, 411-435.
[3] Cockburn, B., Lin, S.Y. and Shu, C.-W. (1989) TVB Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws for Conservation Laws III: One-Dimensional Systems. Journal of Com-putational Physics, 84, 90-113.
[Google Scholar] [CrossRef] 
[4] Cockburn, B., Hou, S. and Shu. C.-W. (1990) The TVB Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws for Conser-vation Laws IV: The Multidimensional Case. Mathematics of Computation, 54, 545-581.
[5] Cockburn, B. and Shu, C.-W. (1998) The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V: Multidimensional Systems. Journal of Computational Physics, 141, 199-224.
[Google Scholar] [CrossRef] 
[6] Aizinger, V. and Dawson, C. (2002) A Discontinuous Galerkin Method for Two-Dimensional Flow and Transport in Shallow Water. Advances in Water Resources, 25, 67-84.
[Google Scholar] [CrossRef] 
[7] Nair, R.D., Thomas, S.J. and Loft, R.D. (2005) A Discontinuous Galerkin Global Shallow Water Model. Monthly Weather Review, 133, 876-888.
[Google Scholar] [CrossRef] 
[8] Schwanenberg, D. and Harms, M. (2004) Discontinuous Galerkin Finite-Element Method for Transcritical Two-Dimensional Shallow Water Flow. Journal of Hydraulic Engineering, 130, 412-421.
[Google Scholar] [CrossRef] 
[9] Fagherazzi, S., Rasetarinera, P., Hussaini, Y.M. and Furbish, D.J. (2004) Numerical Solution of the Dam-Break Problem with a Discontinuous Galerkin Method. Journal of Hydraulic Engineering, 130, 532-539.
[Google Scholar] [CrossRef] 
[10] Eskilsson, C. and Sherwin, S.J. (2004) A Triangular Spec-tral/hp Discontinuous Galerkin Method for Modelling 2D Shallow Water Equations. International Journal for Numerical Methods in Fluids, 45, 605-623.
[Google Scholar] [CrossRef] 
[11] Aureli, F., Maranzoni, A., Mignosa, P. and Ziveri, C.A. (2008) Weighted Surface-Depth Gradient Method for the Numerical Integration of the 2D Shallow Water Equations with Topography. Advances in Water Resources, 31, 962-974.
[Google Scholar] [CrossRef] 
[12] Ern, A., Piperno, S. and Djadel, K. (2008) A Well-Balanced Runge-Kutta Discontinuous Galerkin Method for the Shallow-Water Equations with Flooding and Drying. International Journal for Numerical Methods in Fluids, 58, 1-25.
[Google Scholar] [CrossRef] 
[13] Canestrelli, A., Siviglia, A., Dumbser, M. and Toro, E.F. (2009) Well-Balanced High-Order Centred Schemes for Non-Conservative Hyperbolic Systems. Applications to Shallow Water Equations with Fixed and Mobile Bed. Advances in Water Resources, 32, 834-844.
[Google Scholar] [CrossRef] 
[14] Xing, Y.L., Zhang, X.X. and Shu, C.-W. (2010) Positivity-Preserving High Order Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations. Advances in Water Resources, 33, 1476-1493.
[Google Scholar] [CrossRef] 
[15] Kesserwani, G., Liang, Q., Vazquez, J. and Mosé, R. (2010) Well-Balancing Issues Related to the RKDG2 Scheme for the Shallow Water Equations. International Journal for Numerical Methods in Fluids, 62, 428-448.
[16] Benkhaldoun, F., Elmahi, I. and Sea, M. (2010) A New Finite Volume Method for Flux-Gradient and Source-Term Balancing in Shallow Water Equations. Computer Methods in Applied Mechanics and Engineering, 199, 3224-3335.
[Google Scholar] [CrossRef] 
[17] Kesserwani, G. and Liang, Q.H. (2010) A Discontinuous Galerkin Algorithm for the Two-Dimensional Shallow Water Equations. Computer Methods in Applied Mechanics and Engineering, 199, 3356-3368.
[Google Scholar] [CrossRef] 
[18] Cockburn, B., Karniadakis, G. and Shu, C.-W. (2000) The Development of Discontinuous Galerkin Methods. In: Cockburn, B., Karniadakis, G. and Shu, C.-W., Eds., Discontinuous Galerkin Methods: Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Part I: Overview, Springer, Berlin, 3-50.
[Google Scholar] [CrossRef] 
[19] Shu, C.-W. (2016) High Order WENO and DG Methods for Time-Dependent Convection-Dominated PDEs: A Brief Survey of Several Recent Developments. Journal of Computational Physics, 316, 598-613.
[Google Scholar] [CrossRef] 
[20] Shu, C.-W. (2013) A Brief Survey on Discontinuous Galerkin Ethods in Com-putational Fluid Dynamics. Advances in Mechanics, 43, 541-554.
[21] Toro, E.F., Millington, R.C. and Nejad, L.A.M. (2001) To-wards Very High Order Godunov Schemes. In: Toro, E.F., Ed., Godunov Methods. Theory and Applications, Edited Review, Kluwer Academic Publishers, Dordrecht, 907-940.
[Google Scholar] [CrossRef] 
[22] Dumbse, M., Kaser, M., Titarev, V.A. and Toro, E.F. (2007) Quadra-ture-Free Non-Oscillatory Finite Volume Schemes on Unstructured Meshes for Nonlinear Hypobolic Systems. Journal of Computa-tional Physics, 226, 204-243.
[Google Scholar] [CrossRef] 
[23] Ayaz, F. (2004) Solutions of the System of Differential Equations by Differential Transform Method. Applied Mathematics and Computation, 147, 547-567.
[Google Scholar] [CrossRef] 
[24] Norman, M.R. and Finkel, H. (2012) Multi-Moment ADER-Taylor Methods for Systems of Conservation Laws with Source Terms in One Dimension. Journal of Computational Physics, 231, 6622-6642.
[Google Scholar] [CrossRef] 
[25] Vukovic, S. and Sopta, L. (2002) ENO and WENO Schemes with the Exact Conservation Property for One-Dimensional Shallow Water Equations. Journal of Computational Physics, 179, 593-621.
[Google Scholar] [CrossRef] 

为你推荐